Differential Scanning Calorimeters (DSCs) measure the heat flow to a sample as the sample temperature is varied in a controlled manner. There are two basic types of DSCs, heat flux and power compensation. Brief descriptions of the two types of DSC are included below. A detailed description of the construction and theory of DSCs is disclosed in xe2x80x9cDifferential Scanning Calorimetry an Introduction for Practitionersxe2x80x9d, G. Hxc3x6hne, W. Hemminger and H. -J. Flammersheim (Springer-Verlag, 1996).
Heat flux DSCs include a sensor to measure heat flow to a sample to be analyzed. The sensor has a sample position and a reference position. The sensor is installed in an oven whose temperature is varied dynamically according to a desired temperature program. As the oven is heated or cooled, the temperature difference between the sample and reference positions of the sensor is measured. This temperature difference is assumed to be proportional to the heat flow to the sample.
Power compensation DSCs include a sample and a reference holder installed in a constant temperature enclosure. Each of the holders has a heater and a temperature sensor. The average of the sample and reference holder temperatures is used to control temperature, which follows the desired temperature program. In addition, differential power proportional to the temperature difference between the holders is added to the average power to the sample holder and subtracted from the average power to the reference holder in an effort to reduce the temperature difference between sample and reference holders to zero. The differential power is assumed to be proportional to the sample heat flow and is obtained by measuring the temperature difference between the sample and reference holder. In commercial power compensation DSCs, the difference between sample and reference temperature is generally not zero because a proportional controller is used to control the differential power.
A sample to be analyzed is loaded into a pan and placed on the sample position of the DSC. An inert reference material may be loaded into a pan and placed on the reference position of the DSC, although usually the reference pan is empty. The temperature program for conventional DSCs typically includes combinations of linear temperature ramps and constant temperature segments. Modulated DSC (MDSC) uses a temperature program in which periodic temperature oscillations are superposed on linear ramps and isothermal segments. The experimental result is the sample heat flow versus temperature or time. The heat flow signal is the result of heat flow to or from the sample due to its specific heat and as a result of transitions occurring in the sample.
During the dynamic portion of the DSC experiment, a temperature difference is created between the sample and reference positions of the DSC. In heat flux DSCs, the temperature difference results from the combination of three differential heat flows: the difference between the sample and reference heat flow, the difference between sample and reference sensor heat flow and the difference between sample and reference pan heat flow. In power compensation DSCs, the temperature difference results from the combination of three differential heat flows plus the differential power supplied to the sample holders: the difference between the sample and reference heat flow, the difference between sample and reference holder heat flow and the difference between sample and reference pan heat flow. The heat flow difference between the sample and reference consists of heat flow due to the heat capacity difference between the sample and reference, the heat flow of a transition, or the difference in heating rate that occurs during an MDSC experiment. The heat flow difference between the sample and reference sections of the DSC is the result of thermal resistance and capacitance imbalances in the sensor or between the holders and the difference in heating rate that occurs between the sample and reference sections of the DSC during a transition or during an MDSC experiment. Similarly, the heat flow difference between the sample and reference pans is the result of mass differences between the pans and the difference in heating rate that occurs during a sample transition or during a MDSC experiment.
In conventional heat flux DSCs, the sensor imbalance and pan imbalance are assumed to be insignificant and the differences in heating rates are ignored. In conventional power compensation DSCs, the holder imbalance and pan imbalance are assumed to be insignificant and the differences in heating rates are ignored. When the balance assumptions are satisfied and the sample heating rate is the same as the programmed heating rate, the temperature difference is proportional to the sample heat flow and the differential temperature gives an accurate measure of the sample heat flow. The sample heat flow is only proportional to the measured temperature difference between the sample and reference when the heating rate of the sample and reference are identical, the sensor is perfectly symmetrical, and the pan masses are identical. Proportionality of sample heat flow to temperature difference for a balanced sensor and pans occurs only during portions of the experiment when the instrument is operating at a constant heating rate, the sample is changing temperature at the same rate as the instrument and there are no transitions occurring in the sample. During Modulated DSC experiments, the heating rates of the sample and reference are generally not the same and the difference between measured sample and reference temperatures is not proportional to the sample heat flow.
Thus, the sample heat flow from a conventional DSC is not the actual sample heat flow, but includes the effects of imbalances and differences in heating rates; in other words the DSC sample heat flow measurement is smeared. For many DSC experiments, the smeared sample heat flow is a sufficiently accurate result. For example, when the desired experimental result is the total energy of the transition, such as the heat of fusion of a melt, the total peak area is integrated over a suitable baseline and the result from a conventional DSC is sufficiently accurate. If, however, partial integration of the peak area is required (for example, in the study of reaction kinetics), the smeared sample heat flow of conventional DSC cannot be used. Another example of when the conventional DSC result is inadequate is when two or more transitions in a sample occur within a small temperature interval. In that case, the transitions may be poorly separated in prior art DSCs because of the smearing effects. The improvement in resolution of the present invention greatly improves the separation of closely spaced transitions. In any case, the heat flow signal from prior art DSCs does not accurately portray the sample heat flow during a transition.
During a transition, the heat flow to the sample increases or decreases from the pre-transition value depending upon whether the transition is endothermic or exothermic and whether the DSC is being heated or cooled. The change in sample heat flow causes the heating rate of the sample to be different from that of the DSC and as a consequence, the sample pan and sensor heating rates become different from the programmed heating rate.
U.S. patent applications Ser. Nos. 09/533,949 and 09/643,870, incorporated by reference above, disclose a heat flux DSC that uses a four term heat flow equation to account for sensor imbalances and differences in heating rate between the sample and reference sections of the sensor. The four term DSC heat flow equation derived in the ""949 application is:   q  =            Δ      ⁢              xe2x80x83            ⁢                        T          0                ·                  (                                                    R                r                            -                              R                s                                                                    R                r                            ·                              R                s                                              )                      -                  Δ        ⁢                  xe2x80x83                ⁢        T                    R        r              +                  (                              C            r                    -                      C            s                          )            ·                        ⅆ                      T            s                                    ⅆ          τ                      -                  C        r            ·                                    ⅆ            Δ                    ⁢                      xe2x80x83                    ⁢          T                          ⅆ          τ                    
The first term accounts for the effect of the difference between the sensor sample thermal resistance and the sensor reference thermal resistance. The second term is the conventional DSC heat flow. The third term accounts for the effect of the difference between the sensor sample thermal capacitance and the sensor reference thermal capacitance. The fourth term accounts for the effect of the difference between the heating rates of the sample and reference sides of the DSC.
U.S. patent application Ser. No. 09/643,869, incorporated by reference above, discloses a power compensation DSC that uses a five term heat flow equation to account for sample and reference holder imbalances and differences in heating rate between the sample and reference holders. The five term power compensation DSC heat flow equation derived in the ""869 application is:   q  =            Δ      ⁢              xe2x80x83            ⁢      p        +          Δ      ⁢              xe2x80x83            ⁢                        T          0                ·                  (                                                    R                r                            -                              R                s                                                                    R                r                            ·                              R                s                                              )                      -                  Δ        ⁢                  xe2x80x83                ⁢        T                    R        r              +                  (                              C            r                    -                      C            s                          )            ·                        ⅆ                      T            s                                    ⅆ          τ                      -                  C        r            ·                                    ⅆ            Δ                    ⁢                      xe2x80x83                    ⁢          T                          ⅆ          τ                    
The first term is the difference in power supplied to the sample position versus the power supplied to the reference position. The second term accounts for differences between the thermal resistances of the sample and reference holders. The third term accounts for the heat flow that results from the difference in temperature between the sample and reference. The fourth term is the heat flow resulting from imbalances in thermal capacitance between the sample and reference holders. The fifth term reflects heat flow resulting from differences in heating rate between the sample and reference holders
Heat flow results from the inventions disclosed in the T0 applications show improved dynamic response and hence improved resolution along with improvements in the DSC baseline heat flow.
Rs is the thermal resistance between the sample and the heat source (the xe2x80x9csample resistancexe2x80x9d);
Rr is the thermal resistance between the reference and the heat source (the xe2x80x9creference resistancexe2x80x9d);
Cs, Cr are the thermal capacitances of the sample and reference positions, respectively;
Cps, Cpr are the thermal capacitances of whatever is placed on the sample and reference position; typically, they will be the thermal capacitances of the sample and reference pans, respectively; however, if no pans are used, Cps, Cpr are the thermal capacitances of materials (such as sapphire) having known heat capacity that are placed on the sample and/or reference positions, respectively, without pans;
Rps is the thermal resistance between the sample pan and the sample sensor (the xe2x80x9csample pan resistancexe2x80x9d);
Rpr is the thermal resistance between the reference pan and the reference sensor (the xe2x80x9creference pan resistancexe2x80x9d);
Rc is the thermal resistance between the sample and reference positions;
Rr*=Rr/(1+(Rs+Rr)/Rc) is the composite reference resistance;
Rs*=Rs/(1+(Rs+Rr)/Rc) is the composite sample resistance;
qss is the heat flow to the sample;
q is the differential heat flow to the sample position with respect to the reference position;
Css is the heat capacity of the sample;
Ts is the temperature of the sample position;
Tr is the temperature of the reference position;
Td=Tsxe2x88x92Tr;
Tps is the temperature of the sample pan;
Tpr is the temperature of the reference pan; and
xcfx89 is the angular frequency of the modulation.
In the equations below, a bar above a quantity means its average over one period of modulation, or over integer multiples on one period of modulation. A {circumflex over ( )}over a quantity indicates that it is a complex quantity. Re and Im indicate the real and imaginary parts of a complex quantity, respectively.
The present invention is a method for calibrating the parameters characterizing a DSC cell, including sample and reference pans, and then calculating the heat flow to the sample based upon the results of the calibration. The cell parameters that are calibrated are Rr, Cr, Rsand Cs, and the pan parameters are Rps and Rpr.
A T0 calorimeter is a calorimeter having the structure disclosed in the T0 applications referenced above. The notation used here is essentially the notation used in the T0 applications.
In the absence of any cross-talk, and assuming that the calorimeter can be thought of as having five partsxe2x80x94the base, the two thermocouples, and the two pans (including their contents), and also assuming that the temperature in each part is independent of position and that the heat transfer is between neighboring parts and is proportional to temperature difference, then the model is:                                                         q              ss                        +                                          C                ps                            ⁢                                                ⅆ                                      T                    ps                                                                    ⅆ                  t                                                      +                                          C                s                            ⁢                                                ⅆ                                      T                    s                                                                    ⅆ                  t                                                              =                                    1                              R                s                                      ⁢                          (                                                T                  0                                -                                  T                  s                                            )                                      ,                            (        1        )                                                                    q              ss                        +                                          C                ps                            ⁢                                                ⅆ                                      T                    ps                                                                    ⅆ                  t                                                              =                                    1                              R                ps                                      ⁢                          (                                                T                  s                                -                                  T                  ps                                            )                                      ,                            (        2        )                                                                                    C                pr                            ⁢                                                ⅆ                                      T                    pr                                                                    ⅆ                  t                                                      +                                          C                r                            ⁢                                                ⅆ                                      T                    r                                                                    ⅆ                  t                                                              =                                    1                              R                r                                      ⁢                          (                                                T                  0                                -                                  T                  r                                            )                                      ,                            (        3        )                                                                    C              pr                        ⁢                                          ⅆ                                  T                  pr                                                            ⅆ                t                                              =                                    1                              R                pr                                      ⁢                          (                                                T                  r                                -                                  T                  pr                                            )                                      ,                            (        4        )            
The heat capacity of the sample, Css, is related to the heat flow through       q    ss    =            C      ss        ⁢                            ⅆ                      T            ps                                    ⅆ          t                    .      
Tpr is given, in principle, by (4) so:                                           C            pr                    ⁢                                    ⅆ                              T                pr                                                    ⅆ              t                                      =                              1                          R              pr                                ⁢                                    (                                                T                  r                                -                                                      1                                                                  C                        pr                                            ⁢                                              R                        pr                                                                              ⁢                                                            ∫                      t                                        ⁢                                                                  ⅇ                                                                                                            -                                                              (                                                                  t                                  -                                  τ                                                                )                                                                                      /                                                          C                              pr                                                                                ⁢                                                      R                            pr                                                                                              ⁢                                                                        T                          r                                                ⁡                                                  (                          τ                          )                                                                    ⁢                                              xe2x80x83                                            ⁢                                              ⅆ                        t                                                                                                        )                        .                                              (        5        )            
(The heat capacity Cpr and thermal resistance Rpr are here assumed to change only slowly when compared with the calorimeter-induced transients.)
In the expression for qss, it is assumed that the thermal resistance between the sample and the sample pan is negligible, so that their temperatures can be considered to be equal. Below, the same assumption will be made for the reference side, i.e., the thermal resistance between the reference and the reference pan will be assumed to be negligible, such that their temperatures can be considered to be equal.
The present invention obtains results in a conventional DSC that are quantitatively similar to results obtained using the methods disclosed in the T0 applications. It is also possible to measure all the required quantities (Rr, Rs, Cr, Cs, etc.) through calibration experiments without the need for the T0 thermocouple. Consequently, it is possible, in principle, to obtain the improved baseline and resolution that are the main advantages of the T0 approach without a T0 cell (the issue of cross-talk will be addressed below).
The inventions disclosed in the T0 applications have the advantage that the total calibration procedure is simpler and the reference pan can be changed without the need for recalibration. These are a significant benefits. However, calibration need not be performed often and it is standard practice to leave a reference pan unchanged for long periods of time.
The treatment given here is applied to heat-flux DSCs. Some consideration should also be given to whether there is an equivalent in power-compensation instruments.
In a conventional heat flux calorimeter, the combination Rrxc3x97(3)xe2x88x92Rsxc3x97(1)(i.e., multiply equation (3) by Rr and equation (1) by Rs, and subtract the latter result from the former result) can be used to obtain an expression (see equation 9a below) that is equivalent to the equation used in the ""949 and ""870 applications. For Td=Trxe2x88x92Ts                                           T            d                    +                                    R              r                        ⁢                          C              r                        ⁢                                          ⅆ                                  T                  d                                                            ⅆ                t                                              -                                    (                                                                    R                    s                                    ⁢                                      C                    s                                                  -                                                      R                    r                                    ⁢                                      C                    r                                                              )                        ⁢                                          ⅆ                                  T                  s                                                            ⅆ                t                                                    =                                            R              s                        ⁢                          q              ss                                +                                    R              s                        ⁢                          C              ps                        ⁢                                          ⅆ                                  T                  ps                                                            ⅆ                t                                              -                                    R              r                        ⁢                          C              pr                        ⁢                                          ⅆ                                  T                  pr                                                            ⅆ                t                                                                        (        6        )            
Here, as below, the first term on the left can be considered as equivalent to conventional DSC, the second terms corrects for transient behavior, while the last accounts for asymmetry.